Loop Corrected Supercharges from Holomorphic Anomalies

We describe the loop corrections to supercharges in supersymmetric quantum field theories using the holomorphic twist formalism. We begin by reviewing the relation between supercharge corrections and the "twice-generalized" Konishi anomaly, which corrects the semi-chiral ring. In the holomorphic twist, these corrections appear as BRST anomalies and are computed using the higher operations of an underlying $L_\infty$ conformal algebra. We then apply this formalism to obtain the complete one-loop corrections to the supercharge of four-dimensional Lagrangian supersymmetric gauge theories, including $\mathcal{N}=4$ SYM, where it admits a remarkably compact expression in terms of superfields.